Why does pressing a transform hotkey not display the axes on the selected object?
That is by design. When you are using shortcuts you can restrict the movement to a single axis by pressing G followed by X, Y or Z. Alternatively to restrict the motion to only two you can use SHIFT + the axis you don't want any movement on. Therefore there is no need for the gizmo in that context.

If you want to enable the move version with the gizmo press SHIFT+SPACEBAR+G or you can right click on the icon in the toolbar and assign your own shortcut. Just make sure it doesn't conflict with any existing shortcuts.

The gizmos for translate, rotate and scale can be permanently enabled in the Viewport Gizmos.